Low-Density Ferromagnetism in the Hubbard Model

We study analytically the stability of the saturated ferromagnetic ground state (the Nagaoka state) of the Hubbard model in the low electron density limit with the mean-field approach. We find that the critical U_c is just 4πt, where t is the hopping integral between the nearest neighbours, i.e., when U is larger than 4πt,the paramagnetic state is unstable with respect to the ferromagnetic state, and vice versa.